Signatures in TQFT : Asymptotics and Modularity

We study the signature ${\sigma }_g(\frac{q}{p})$ of ${\mathrm{S}\mathrm{U}}_2$-TQFT vector spaces associated to surfaces of genus $g$, as a function of the defining root of unity $\zeta =e^{i\pi q\mathrm{/}p}$. We prove that $\frac{1}{p^2}{\sigma }_2(\frac{q}{p})$ converges to $\mathrm{\Lambda }(\theta )=\frac{16}{{\pi }^3}\sum _{n\ge 1,\text{ odd}}\frac{1}{n^3\sin (n\pi \theta )}$ when $\frac{q}{p}$ goes to an irrational number $\theta \in [0,1]$ under certain conditions. We also observe that the function $\mathrm{\Lambda }(\theta )$ is the boundary value of an Eichler integral of a level $2$ modular form of weight $4$, and use this to propose a conjectural transformation law for the signature function in genus 2 similar to the reciprocity formula for classical Dedekind sums.